nLab subsingleton

Definition

A subsingleton generally refers to a subset (of some ambient set $A$) having at most one element. That is, it is a subset $B$ of $A$ such that any two elements of $B$ are equal.

Of course, classically any subsingleton is either empty or a singleton, but constructively this need not hold. In a topos, the “object of subsingletons in $A$” is the partial map classifier for $A$, often denoted $A_\bot$.

Remark

Sometimes a slightly different convention is used: There what we call subsingletons are called subterminals, and a subset $B$ of $A$ is a subsingleton if and only if there exists an element $a \in A$ such that every element of $B$ is equal to $a$. With this nomenclature, any subsingleton is a subterminal, but the converse doesn’t hold in general. (See flabby sheaf for a class of examples where the converse does hold.)

 Generating subsingletons from propositions

Given a proposition $P$ and a singleton $\{*\}$, one can construct the subsingleton

$$\{p \in \{*\} \vert (p = *) \wedge P\}$$

This means that one can use the set-theoretic operations to define the BHK interpretation of logic: Given two propositions $P$ and $Q$, purely $P$ or $Q$ is given by an element of the binary disjoint union

$$\{p \in \{*\} \vert (p = *) \wedge P\} \uplus \{p \in \{*\} \vert (p = *) \wedge Q\}$$

and the pure existential quantifier of a predicate $P(x)$ on a set $A$ is given by an element of the indexed disjoint union

$$\biguplus_{x \in A} \{p \in \{*\} \vert (p = *) \wedge P(x)\}$$

Relation to the principles of omniscience

Suppose that $A$ is a subsingleton. Then the existential quantifier $\exists x.x \in A$ which says that $A$ is an inhabited set is equivalent to the existential quantifier $\exists x:A.(\lambda t.1)(x) = 1$, where $\lambda t.1$ is the constant function from $A$ to the boolean domain which takes elements of $A$ to the boolean $1 \in \mathrm{bool}$. Then, one can show that various principles of omniscience are equivalent to weak versions of the internal excluded middle, via the special case of the constant function $\lambda t.1$:

• That the limited principle of omniscience holds for all subsingletons is equivalent to internal excluded middle.

• That the weak limited principle of omniscience holds for all subsingletons is equivalent to internal weak excluded middle.

• That the lesser limited principle of omniscience holds for all subsingletons is equivalent to internal de Morgan's law.

• That Markov's principle holds for all subsingletons is equivalent to the internal double negation law.

See also

• h-proposition
• support
• decidable subsingleton

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language